The FBSDE approach to sine-Gordon up to 6π

We develop a stochastic analysis of the sine-Gordon Euclidean quantum field (cos (βφ))_2 on the full space up to the second threshold, i.e. for β^2 < 6 π. The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition (X_t)_t ⩾ 0 of the interacting Euclidean field X_∞ along a scale parameter t ⩾ 0. This FBSDE describes the optimiser of the stochastic control representation of the Euclidean QFT introduced by Barashkov and one of the authors. We show that the FBSDE provides a description of the interacting field without cut-offs and that it can be used effectively to study the sine-Gordon measure to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties.